On critical $\mathrm{L}^p$-differentiability of $\mathrm{BD}$-maps

TL;DR

This paper establishes that functions of locally bounded deformation are almost everywhere differentiable in the critical L^{n/(n-1)} sense, extending to functions of bounded A-variation with finite dimensional null-space.

Contribution

It proves critical L^p-differentiability for functions of bounded deformation and bounded A-variation under specific conditions, generalizing previous results.

Findings

Functions of bounded deformation are L^{n/(n-1)}-differentiable almost everywhere.

The result extends to functions of bounded A-variation with finite dimensional null-space.

Critical L^p-differentiability holds for a broad class of functions with linear differential operators.

Abstract

We prove that functions of locally bounded deformation on $\mathbb{R}^n$ are $\mathrm{L}^{n/(n-1)}$-differentiable almost everywhere. More generally, we show that this critical $\mathrm{L}^p$-differentiability result holds for functions of locally bounded $\mathbb{A}$-variation, provided that the first order, homogeneous, linear differential operator $\mathbb{A}$ has finite dimensional null-space.